Binary and Hexadecimal numbers

ICT Foundation

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Copyright © 2010, IT Gatekeeper Project – Ohiwa Lab. All rights reserved.


Binary and Hexadecimal numbers

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Computer and binary numbers

• Inside the computer, the binary string (bit string) is used for information processing▪ Based on the use of high and low voltages to

process information, using the binary numbers is easier than using the decimal numbers to make a calculation

▪ To calculate decimal numbers, first, they are converted to binary numbers. The binary numbers are then calculated. Finally, the resulting binary numbers are reconverted to decimal numbers for displaying.

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Binary number

• Using 0 and 1

Decimal number

0 1 2 3 4 5 6 7

Binary Binary numbernumber 00 11 1010 1111 100100 101101 110110 111111

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Radix (or Base)

• Radix of Decimal number is 10▪207 （ 10 ）＝ 2×102 ＋ 0×101 ＋ 7×100

• The first digit is 100 （＝ 1 ）• The second digit is 101

• The third digit is 102

• Radix of binary number is 2▪101 （ 2 ）＝ 1×22 ＋ 0×21 ＋ 1×20

• The first digit is 20 （＝ 1)• The second digit is 21

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Using binary number to represent the weather

• One card has front as 0 and back as 1 value.

• To represent 4 types of weather, 2-digit binary number is needed.

sunny 00

rainy 01

snowy 10

cloudy 11

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The relation between thebinary number and the amount of information

• One-digit binary number▪ distinguishes 2 kinds of information

• Two-digit binary number▪ distinguishes 4 kinds of information

• Three-digit binary number▪ distinguishes 8 kinds of information

We will study about the amount of information in the 7th lecture

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Conversion from binary to decimal number

• Calculate the sum of the corresponding power two of weight value for all digits 1 in binary number. (the weight of the left-most digit is 0, the next one is 1…)

1

26

0

25

0

24

1

23

1

22

1

21

1

20

1

27

11001111 （ 2 ）＝ 128 ＋ 64 ＋ 0 ＋ 0 ＋ 8 ＋ 4 ＋ 2 ＋ 1 ＝ 207 （ 10 ）

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Conversion from decimal to binary(Integers)

• Arrange in reverse order of modulo 2

2 ） 207 2 ） 103 2 ） 51 2 ） 25 2 ） 12 2 ） 6 2 ） 3 2 ） 1 　　 0

207 （ 10 ）＝ 11001111 （ 2 ）

Arrange in reverse order

Modulo 2

・・・ 1・・・ 1・・・ 1・・・ 1・・・ 0・・・ 0・・・ 1・・・ 1

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Conversion from decimal to binary(Fraction)

• Arrange the integer part when multiplying fraction with 2

0.625× 2 1.25

0.25× 2 0.5

0.5× 2 1.0

0.625 （ 10 ）＝ 0.101 （ 2 ）

arrange

Integer part when

multiplying with 2

・・・ 1

・・・ 0

・・・ 1

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Hexadecimal number

• It’s hard to read the binary number once its number of digits increases.

• One digit of hexadecimal number is combined by 4 digits of binary number.

Decimal number

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Hexadecimal Hexadecimal numbernumber 00 11 22 33 44 55 66 77 88 99 AA BB CC DD EE FF

Used for character coding (lecture 5), displaying colors in web page (lecture 9), programming, etc.

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Conversion from binary to hexadecimal

number• From the left-most digit, divide to groups

that contains each four digits.

• Convert each 4-digit binary number group to decimal number, and then convert it to hexadecimal.

100

4

4

1110

14

E

0110

6

6

10011100110 （ 2 ）＝ 4E6（ 16 ）

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Conversion from hexadecimal to decimal

number• Calculate the sum of (weight of each digit *

digit value)

• The left-most digit is 160, the next one is 161…

4E6 （ 16 ）＝1254 （ 10 ）

4

162

E （ =14 ）

161

6

160

4E6 （ 16 ）＝ 4×256 　＋　 14×16 　＋　 6×1 　＝ 1254 （ 10 ）

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Binary and Hexadecimal numbers